THE PSEUDO-ATOM : A SOLUBLE MANY BODY
PROBLEM
M. Moshinsky, O. Novaro, A. Calles
To cite this version:
M. Moshinsky, O. Novaro, A. Calles. THE PSEUDO-ATOM : A SOLUBLE MANY
BODY PROBLEM. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-125-C4-140.
<10.1051/jphyscol:1970422>. <jpa-00213877>
HAL Id: jpa-00213877
https://hal.archives-ouvertes.fr/jpa-00213877
Submitted on 1 Jan 1970
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JOURNAL DE PH\'SIQUF
Collorlirc C4. .\icl~plbtjic,~~t
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tio
1 1 - 12. T O I I I 3L1J. I\'oI..-DL?. 1970. pa+y C3- 125
THE PSEUDO-ATOM : A SOLUBLE MANY BODY PROBLEM (*)
M. MOSHINSKY, 0 . NOVARO
(t) and A. CALLES (++)
Instituto de Fisica, Universidad d e MCxico, Mtxico, D. F.
Resume. - Si on neglige tous les effets relativistes, un atonie ou plus generalement un ion de
charge Z, est un systeme de 11 fermions de spin t qui sont dans un potentiel coulombien attractif
et interagissent par l'inter~iiediaire d'une force de Coulomb repulsive. Conime ce probleme
n'a pas de solution exacte pour 11 > I . Lln grand nonibre de techniques d'approximation ont Cte
employees dans son etude et, dans beaucoup de cas, sans qu'il soit fait une analyse critique suffisante. Dans cet article on remplace le potentiel coulombien conmun par un oscillateur harmonique
attractif et la force d'interaction par un oscillateur harmonique repulsif. Le modele qui en resulte,
que nous appelons pseudo-atone, peut Ctre resolu exactement pour n'iniporte quelle valeur de n.
On applique alors a ce probleme des procedes d'approxiniation bases sur les niethodes de perturbation, de variation, de Hartree-Fock, de la variance, de la niatrice densite, etc ... On discute,
dans toutes ces methodes, la convergence vers les v a l e ~ ~ rexactes
s
de I'energie et de la fonction
d'onde, tout d'abord pour 11 = 2. electrons, ensuite pour des pseudo-atomes a couches conipl&tes,
et ceci jusqii'a I 1 2 electrons.
Abstract. - If we disregard a n y relativistic effects, an atom, or more generally an ion of
charge Z, is a system of 11 fei-mions of spin whicli are in an attractive Coiilonib potential and
interact through a repulsive Coulomb force. As this problem is not exactly s o l ~ ~ bfor
l e any 11 > 1,
a large number of approximation tecliniqi~eshave been used in its discussion in many cases without
sufficient critical analysis. In the present paper we replace tlie Coillomb c o ~ i i ~ ~ potential
ion
by an
attractive harmonic oscillator and tlie interacting force by a repulsive liarriionic oscillator. The
resulting model whicli we call a pseudo-atom can be solved exactly for any 11. We then proceed
to apply to this problem approxi~iiationprocedures such as those based on perturbation, variation,
Hartree-Fock, variance, density matrix, etc. metliods. We discuss the convergence of all these
methods to the exact energy and wave function, first for 11 = 2 electrons and then for closed shell
pseudo-atoms that go up to 112 electrons.
1. Introduction. - If we disregard any relativis~ic
effects, a n atom, o r more generally an ion of charge
Z, is a system of ir fermions of spin ?, whose hamiltonian, in atomic units, is given by
was available, led to systematic procedures for
determining the correlation effects. In particular
Sinnnoglu [ 5 ] , among others, developed procedures
for dealing \vitIi these correlation phenomena in
a perturbation o r variational way but starting with
11
'I
I
+
11
HF states. Other variational procedures based on
j < j = 2 1 rj - rj 1 .
the variance 161, rather than on the expectation value
of the liamiltonian, liave been used by Conroy [7].
AS is well known, tlie eigenvalues and eigenstates
The density matrix [8] has also played an important
of (1. I ) cannot be obtained exactly for any 11 > 1.
role in the attacks or1 atomic problems and, in partiThus from tlie very beginning of the applications
cular. the concept of natural orbitals derived fro111
of quantum mechanics to atoms of more than one
it by Lowdin [9] has proved both conceptually and
electron, use was made of approxiniation t e c h n i q ~ ~ e s
numerically fi-uitt'~11.
whose degree of sophistication has been increasing
Wliile tliese and other approxirnntion procedures
with time. Initially one was satisfied ivitli first o r
have been discussed at great length, their critical
second order perturbation calculations [I] 01- simple
analysis has lagged behind their systematic application.
variational analysis [ 2 ] . Later tlie metllods developed
Thus sometimes tlie results of tliese procedures liave
by Hartree [3] and their antisymmetrized version [4]
questionable validity.
( H a r t r e e - ~ o c k(HF)) came into vogue and are still
This situation has led us t o investigate many body
much in use today. The interest in tlie residual part
problems in wliicli tlie common potential and interof the energy a n d wave fi~nction,once the HF solution
acting- force liave a simple form for wliicli exact
solutions are available. We could then apply the
(*) Work supported by Comisi6n Nacional de Energia
approximation techniques of atomic physics to tliese
Nuclear (Mexico).
re dictions
(t) Research Associate of lnstituto Mexican0 dcl Petr61eo. many body problems and test their
against the exact results. Specifically, the harmonic
(tt) Graduate Fellow of the Ford Foundation.
1
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970422
M. MOSHINSKY, 0 . NOVARO AND A. CALLES
C4-126
oscillator potential is one such simple interaction
and if we replace the Coulomb potential by it in (1.1)
I. e.
we get the hamiltonian of what we shall denote as
a pseudo atom
If we introduce the n
by the definition
- 1 relative Jacobi coordinates
and the center of mass by
the hamiltonian (1.1) takes the form
Looking at the hamiltonian (1.5) from a classical
viewpoint, we see that the restoring force for each
normal coordinate is proportional to - I': with the
coefficient being Z - 11 for i = 1, ..., 11 - 1) and Z
for i = 11. Thus the system will be clearly unstable
for Z < 11, and for Z = n we have free particle
.
motion ill all the relative coordinates ( 1 . 4 ~ )Therefore
we will only be interested in problems in which
Z > ir. We shall denote pseudo-atoms of charge Z
by tlie corresponding cliemical symbol preceeded
by the letter P and with as many upper indeces + as
tlie degree Z - it of ionization. Thus, for example,
P L i f is tlie pseudo-atom of charge Z = 3 and number
of electrons it = 2.
The energy levels of the hamiltonian (1.5) are
given by
FIG. 1. - Energy levels for the pseudo-atom characterized by
the quantum numbers (N', N) in (1.6) with Z = 2 n .
dinates vf. In the following sections we shall give
tlie explicit eigenstates in a number of particular cases
as well as sketch the procedure to be followed in
general.
The simplest many body pseudo-atom is the one
that has n = 2 electrons. As many of tlie usual
approximation techniques of atomic theory are
already applicable in this case, we discuss it in detail
in the paper both because of its intrinsic interest
and the insight it provides for the 11-electron pseudoatom.
2. The two electron pseudo-atom. - Before we
proceed to discuss tlie exact and approximate solutions of our problem we introduce tlie notation used
in the present article for harmonic oscillator states.
The single particle liarmonic oscillator state correspollding to tlie hamiltonian
will be given in any one of tlie following notations
1/9,,~,~(o,
I') = I
where N is tlie number of quanta associated with
the states in the 1t - 1 relative coordinates, while N'
is tlie number of quanta of the state associated with
the center of mass coordinate. For Z = 2 11 tlie energy
levels are plotted in figure 1 in which the energy scale
is given in units of (2 TI)%.
TIie eigenstates of the hamiltonian (1.5) and of
the other possible integrals of motion of the problem,
are more difficult to determine as we must also satisfy
tlie Pauli principle. While the antisymmetrization
of the wave function is simple in terms of the coordinates vi and their corresponding spins, it is more
complicated when we introduce the normal coor-
W , 111111
>=
= o3I4 ~,,,(o"'r)Y
,,,,(0, rp),
(2.2)
where Y,,,, is a spherical harmonic and R,,,(r) tlie radial
wave function as defined in reference [lo]. When
the frequency o = 1, we shall suppress tlie o everywhere it appears, i. e.
When dealing with a two particle harmonic oscillator state in which tlie first has frequency o,,and
the second 02,
the state of total angular momentum L
and projection M will be given in any one of the following notations
THE PSEUIIO-ATOIM : A SOLUBLE MANY BODY PROBLEb1
CJ-127
In particular, the ground state of our system has
= I; = n' = 1; = 0 and thus is a singlet whose
wave function has the form
11;
where the square bracket stands for vector coupling
of the angular momenta I,, I, to L. When both
o,= o2= 1 we shall suppress thc frequencies
in the Y and the kct.
When instcad of the coordinates r , , r,, we use the
Jacobi coordinates r ; , r i for thc two particle system,
we sl~alldenote thc wave functions with primes $',
Y' and the kets with round brackets. Also, whcn
possible, the quantum numbers will be primed.
Thus for cxamplc, for o , = to, = 1 we have
Y~ililliliL~,,,.(r~,
r;) =
1
ti',
l',
11;
1; L' M')=
and its energy is
Eb
=
.2[(1 - 2 6)"
+ 1] .
(2.13)
We proceed to expand both YJ; and Eb in a power
series in (5, which will allow us to carry an infinite
ordcr perturbation analysis of our problem starting
from the non-interacting states.
h) THEPERTURBATION
APPROACH.
- If we consider
a) EXAC-rSOLUTIONOF THE Two ELI:CI.R~N
SYSthe
interaction
in
(2.7)
as
a
perturbation
with 6
TEM. - For 11 = 2 it is convenient to carry out a
the
perturbation
expansion
being
its
parameter,
scale transformation in cquation (1 .3) of the form
rcquires the development
Eb = E o + E 1 6 + . . . + E p 6 P + . . . ,
which will give us the hamiltonian
f,'
=
+
(I - 2 3;) r ; 2 ] f [P>2+ I.;?] , (2.70)
where
5 =z-I.
(2.76)
The advantage of this scale transformation is that
in the non-interaction part of the hamiltonian the frequency is now w = 1 and thus we avoid bothersome
factors in any expansion of the exact solution in terms
of the non-interacting states.
The eigenstates of (2.7) o f total orbital angular
momentum L and projection M are given by
and their corresponding energies, which are independent of L, are
As in the case
given by
11 =
2 the primcd coordinates are
it is clear that the states with even (odd) I; are symmetric (antisymmetric) under exchange of the coordinates
r , and r,. Thus the Pauli principle requires that :
States with even 1; are singlet ( S = O),
States with odd 1; are triplct ( S
is the total spin.
=
(2.1 1)
I), where S
=
A(Qo
+
6
+ ... + cPP SP + '.'),
(2.14~)
(2.14b)
wherc the coefficient A is selected so as to guarantee
that @, is normalized. Both Ep and Q1, are independent
of s .
From (2.13) we immediately obtain that
The determination of @, requires a little more
thought. We first notice we could expand
where [lo]
x exp
[- 5 (I
1
- 2 6)"'
]
r2] Rno(r) r 2 d r
where r is a gamma function. For n = 0,a, reduces
to the first factor in (2.17) and thus from (2.12)
we see that the coefficient A in the expansion (2.146)
is given by
M. MOSHINSKY, 0. NOVARO A N D A. CALLES
C4-128
Replacing now (2.16)in (2.12) and using the development
we see that in the expansion of A-I Yb, @, is given by
the variational analysis i~nplies that we calculate
the matrix of the hamiltonian Je of (2.7) with respect
to these states and proceed to diagoiialize it to obtain
the energy values and eigenstates, the latter as linear
combinations of (2.25). As the matrix is infinite,
we have to cut it somewhere and a reasonable way
would be to cut it at a given maximum number
of quanta which we denote by p. The positive parity
of the ground state indicates that p must be limited
to even values. The set of states (2.25) we consider
is then restricted by
This implies that equivalently we can work in the
primed system of coordinates r ; , r; where tlie corresponding states (2.21) will be restricted by
with
n'
(2 p - 1) !
x -.
2'-' (p + n') ! (p - n') !
(2.20b)
Thus we have the explicit form of @, as a function
of the coordinates v;, u;. Had we wanted the @,
in terms of the original coordinates u,, r,, we just
have to develop round bracket states in terms of
angular ones, e. g.
where < ( > are tlie well known
harmonic oscillator
[ll].
In
tlie
present problem
transformation brackets
we have even tlie further restriction 0; = 0, but we
put the more general expansion to include also
the Hartree-Fock case to be discussed below.
We are now in a position to compare the results
of the exact and perturbation calculations both in
regard to energies and to the overlap of the wave
functions. For tliis purpose we define
Eb
and
1-
/
Diagonalizing the matrix for 2 n;, 2 M; < p we obtain,
among others, the energy and normalized wave
function for the ground state. We can then, as in
(2.23), compare them with the exact solution, which
we will do in section i). It is important to note that
our variational ground state, as the perturbational
one of the previous subsection, is given in terms
of tlie primed coordinates r ; , r ; . We c o ~ ~ transform
ld
it to ordinary coordinates with the help of the brackets
in (2.21).
d) THE HARTREE-FOCK
(HF) APPROXIMATION.
The hamiltonian (1.3) for a two particle system has
the form
We can then plot, as a f ~ ~ n c t i oofn y , both
6, - E;,
From (2.20) we see that all kets required in tlie present
problem have 11; = 0 and thus we need only the
matrix elements [12]
).P) 12, (2,230, b)
(xPjxP)
(yA5
where the round brackets stand for the scalar products and
A discussion of numerical results for up to p = 10 will
be given in Section i) of this chapter.
c) THEVARIATIONAL
APPROACH.
- If we start from
the non-interacting states in the variables u,, u2
of zero total orbital angular momentum, i. e.
In, I n 2 1 0 0 >
(2.25)
As the ground state is a singlet, and thus anti-symmetric in the spins, we can propose for the configuration
space part of the HF solution of (2.29) two particles
in the same orbital i. e.
The standard analysis [13] gives, for the single particle
state $(r,), the equation
T H E PSEUDO-ATOM : A SOLUBLE M A N Y BOD\- PROBLEM
This equation appears to be integrodifferential, but
actually, as $(r2) is of definite parity, the expectation
value of the term r , .I-, vanishes and thus it becomes
+ (Z - 1)
i-f]
$ ( r l ) = e' $ ( r l )
(2.33)
Clearly then the H F average potential is also of the
harmonic oscillator type but of frequency (Z - 1)'
rather than the Z' of the one body part of the hamiltonian (2.29). From (2.33) the energy levels in this
potential are given by
= (Z
- i ( Z - I)-' (2 12
- 1)% ( Z - -:-) ( Z - I)-' (2 11 + I
E'o - +[(1 -
+ 1 + 311
+ 3) ,
(2.34)
x exp[-
-
E - 3 E0 -
We can then use these states to carry out a perturbational or variational analysis for the exact problem.
Before proceeding with this program we carry out
a scale transformation
1, 2 ,
=
M;',I;?
Y
+ +[pi2+ (1 +E ) ri2] ,
(2.39)
(Z -
(2.40)
E) I ' ; ~ ]
where
E =
])-I.
3 (2p - 3)!!
2"
p!
,
= - -
p
k &)%, r )
(2.44)
3
We shall use tlie liamiltonian
in the following
sections, as in tliis way we avoid botliersome factors
in any expansion of the exact solution in terms of HF
states.
1 1 (-
~)1"+''2
rllY2
+ 2 9 , - I ) ! (212; + 2 9 , - I ) !
q 1 ! q2 ! ( 2 n; + q,) ! (2 n; + q,) !
(2n;
with q,, 9 , restricted by
q1
+ (1 -
. (2.42)
where
(2.38)
on tlie HF wave functions (2.37), so that they become
states of frequency w = 1 . The same transformation
applied to the hamiltonian (2.29) will give us
=
+ &)'I2
in terms of $,,oo(r). This can be done as in (2.16)
and combining terms we can expand Pb in a power
series where the coefficient of E,, which we designate
by &,,, take the form
Once we have the HF common potential for tlie two
particle system, the corresponding complete family
of eigenstates is given by
=
r;*] x
E)"'
It is interesting to note that in this case only even
orders of the perturbation contribute to the energy.
For the eigenstate (2.42) we need first expand
(2.36)
ri 4 (2- 1)-'I4 ri , i
3(1
(2.41)
To carry out the perturbation approach with H F
states, we must develop Eh and !Pb in a power series
in E similar to tlie development (2.14) in 6. For the
energy tliis is trivial and we get for the coefficient
of &,,
we designate by Ep,
values
$ooo((l
The HF energy of tlie ground state is given by the
expectation value of H of (2.29) with respect to the
state (2.30), in which $ ( r l ) is the lowest eigenstate
of (2.32). As the expectation value of r , .r2 is again
zero, we obtain
+(I -
Lf/; = T C - ~ /-~ E( L
)-7 ' 8 exp[-
and the corresponding eigenstates become
(2.35)
+ (1 + & ) ' I 2 ] ,
&)I;'
-.
(2.32)
where
-
-
C4- 129
As in
+ q2 + 11; + n;
=
p , q1 ,< p, 92
<P.
now define
b, we
-
-
Ep = Eo + E , E + ... + E
~ E ~ ,
and plot as function of p both the energy ratios and
overlaps with the exact solution that were defined
in (2.23). Numerical results for up to p = 10 will
be given in section i).
Again we note that our perturbational wave functions are expressed in terms of the 0, of ( 2 . 4 9 , which
depend on the coordinates r ; , r;. If we want toexpress
them in terms of states depending on r,, r, we must
use the transformation brackets in (2.21).
e) THEPERTURBATION
APPROACH
WITH H F STATES.
- The ground state energy and wave function of
the hamiltonian
of (2.39) are given by
f ) THEVARIATIONAL
APPROACH
WITH HF STATES.
As in section c) the variational approach require the
C4-130
M. MOSHINSKY, 0 . NOVARO A N D A. CALLES
calcglation of the matrix of
with respect to the
states (2.25). If we limit the matrix to agiven maximum
number of quanta that we denote by p, it is equiva-
lent to considering the set of states (2.21) in which
2(n;
+ n;) 6 p .
(2.48)
We thus need only to calculate the matrix elements
(ii; 0
n; 0
00 1
XI n; 0 n; 0 00) = (2 11; + 2 ni + 3) d;;,;
Diagonalizing the matrix in which (2.48) holds for
both n;, n; and ii;, n; gives us, among others, the
ground state energy and normalized wave function,
the latter in terms of HF states. Numerical results will
be discussed in section i).
g) THE MINIMIZATION
OF THE VARIANCE.
- SO far
we have discussed only variational approaches based
on the expectation value of the hamiltonian using
either non-interacting or HF complete sets of two
particle states. Another possibility is the variational
approach based on the variance [6, 71 i. e. the expectation value of the operator
where H is the hamiltonian of our problem (in the
present case (2.29)) and E the ground state energy
(in the present case zli Eh or ( 2 - 1)
depending
on whether we use non-interacting or HF states).
As H is hermitian, the operator (2.50) has clearly
only non-negative eigenvalues, and if we are able to
find the exact eigenstates one of them would be the
ground state wave function of our problem corresponding to the eigenvalue zero of (H - E ) ~ .
We could now carry out a variational analysis
for (H - E)2 starting either from the non-interacting
or HF states for u p to p quanta. The matrix elements
of (H - I?)' with respect to these states can be immediately obtained either from (2.28) or (2.49) and thus
the eigenvalues and eigenstates of the operator (2.50)
can be derived. The lowest eigenvalue gives the
minimum of the variance and it clearly should approach
zero when p -,co.The ground state energy associated
with the minimum variance can be obtained by
considering the expectation value of the hamiltonian
with respect to the corresponding eigenstate of
(H - E)'. We can compare the variance results for
ground state energy and wave function, which are a
function of p, with the exact solution. This will be done
in section i) for up to p = 10 quanta.
In an actual physical problem the variance analysis
is implemented by substituting for E in (2.50), the
experimental value of the ground state energy.
1/2&
-
bounds for the ground state energy. It is very convenient to be able to estimate lower bounds also, as then
we can bracket the exact energy in a narrowing
region as we increase the degree of approximation.
Many lower bound criteria exist. Here we sl~all
only mention a criterium developed by Temple [I41
as modified by Conroy [7], which requires the knowledge of the energy G, and the variance for the corresponding wave function Y, of any type of variational
solution that gives upper bounds, plus the experimental energy E, of the first excited state accessible to the
same variational calculation. The lower bound is
then given by the formula
where U: is the variance, i. e. the expectation value of
the operator (2.50) with respect to Y', and we designate by E, the exact energy of the ground state.
If we take Y, to be the wave function corresponding
to the minimum value of the variance, then the value
of U: for a variational analysis going up to p quanta
was obtained in the previous section and so we could
immediately determine the lower bound. Numerical
results will be discussed in the next section.
i) NUMERICAL
RESULTSFOR THE DIFFERENT
APPRO-
METHODS.
- We proceed now t o discuss
numerical results for PLi' when we apply the different
approximation methods we proposed.
Let us first consider the perturbation procedures
both with the non interacting and HF states. In figure 2
we plot the energy deviation from the exact value
(defined in (2.230)) as a function of the order of the
perturbation p. The deviation is given i n a logarithmic
scale that goes from 1 to lo-". We see that the
perturbation series converges rapidly but much more
so for HF than for non-interacting states. The same
holds for the deviation from 1 of the overlap of the
exact and approximate wave functions given by (2.236),
as seen in figure 3. In fact, the same order of approximation seems to be achieved in 4th order perturbation theory with HF states as in 8th order for noninteracting states.
17) UPPER A N D LOWERBOUNDSFOR THE GROUND In figure 4 we consider the energy deviations as
- Variational analysis provide upper Function of the number of quanta p for the variational
STATEENERGY.
XIMATION
T H E PSEUDO-ATOM : A SOLUBLE MANY BODY PROBLEM
C4-131
I.
I PERTURBATIONS HF
PPERTURBATIONS STANDARD
IVARIATIONAL HARTREROCK
2VARlATlDNAL STANDARD
3.VARIANCE H F
4.VARIANCE STANDARD
ORDER OF PERTURBATION
-
P
P
NUMBER OF QUANTA
"
"
0
4
6
8
9
;
o
FIG. 2. - Deviation ratio of the pth order perturbation energy
FIG. 4. - Deviation ratio of the variational and variance
from the exact value for PLi+. The deviation in this and the
following three figures are given in a logarithmic scale for
both HF and non-interacting states.
energies for up to p quanta from the exact value for PLif.
I
NUMBER OF QUANTA
I
ORDER OF PERTURBATION
't
I
FIG. 3. -Deviation
and
from 1 of the overlap between the exact
Pfl1 order perturbation wave functions for PLif.
and variance procedures both witli non-interacting
and H F states and in figure 5 we d o the same for the
deviation of the overlap from 1. Again tlie use of H F
states improves matters considerably as, for example,
we get the same degree of approximation when we go
up to 6 quanta in the variational approximatioli
witli H F states, as when we go u p to 10 quanta witli
ordinary states. Results for tlie variance analysis
differ very little from those of the variational one when
we use the same states and number of quanta, though
the variational is slightly better.
We complement figure 4 witli Tables I and 11. I n
Table I we give, for non interacting states, the upper
bounds derived from variational and variance calculations, as well as the lower bound of (2.51) as function of the number of quanta. We also give the
variance minimum, i. e. tlie lowest eigenvalue of
(H - E)' as a function of p. In Table I1 we d o tlie
same thing for H F states. Note that the limiting values
0
1
2
3
4
5
6
P
7
8
9
1
0
FIG. 5. - Deviation from I of the overlap between the exact
and the variational and variance wave functions of PLi+ for
up to p quanta.
of the energy, when p increases, d o not agree in
Tables I and 11. This is due t o the fact that the exact
values are EA and EA which are related by
ZS
EA
=
( Z - I)%
EA
with Z
=
3
for PL;
Upper (uariational and variance) and lower boutfds
of' the energy and the lninirnum value of the variance
for PL~' as function of the number of quanta.
The non-inferacfion states were used
Number
of
quanta
0
2
4
6
8
10
Energy
(variational)
2.500 001
2.384 941
2.368 395
2.366 285
2.366 052
2.366 029
Variance
0.184 615
0.034 194
0.006 330
0.000 973
0.000 126
0.000 01 5
Energy
(from
variance)
2.500 001
2.388 617
2.369 076
2.366 351
2.366 057
2.366 029
Lower
bound
2.319 134
2.354 825
2.362 908
2 365 442
2.365 943
2.366 01 6
M . bIOSI4INSKY. 0. NOVAKO AND A. CAI-LES
and the two particle density matrix has the form
Uppo (cariational arzcl cariatzce) atit/ 101t.o bo~1tid.v
of' tllc energj. an(/ the tiiiiiitiilitii rallte oj' the cariailce
/or PI,;'
as firtictiotz of' tllv rlltiiiber of' qt~atzta. where the primed coordinates in the wave functions
are expressed in terms of the ordinary ones through
Tlle HF slates ,+.ere used
0
2
4
6
8
10
Energy
(variational)
3.000000
2.905 091
2.898 177
2.897 796
2.897 778
2.897 778
Variance
-
0.197 942
0.021 023
0.001 734
0.000 107
0.000 005
0.000 000
Energy
(from
variance)
3.000 000
2.906 400
2.898 245
2.897 798
2.897 778
2.897 778
Lower
bound
-
2.849 128
2.890 148
2.896 95 1
2.897 720
2.837 774
2.897 778
Again we get a quicker narrowing of the gap
between upper and lower bounds, and a faster convergence 01' the variance minimum to zero. when we
use H F rather than non-interacting states. The
convergence of the lower bound is quite satisfactory
though it reaches the exact energy less pronlptly
than the Ritz upper hound.
the relations (2. 10).
The more interesting one body density matrix is
obtained tl~rough the definition
which, carrying out the elementary integrations, has
the explicit form
where, for later notatio~zal convenience, we liavc
i., by r,. The last factor in ( 2 . 5 7 ) shows
clearly the correlation effect due to the interaction,
as it cannot be written as a product of independent
functions of r 1 and r,.
The density matrix ( 2 . 5 7 ) has two c o n t i n u o u ~
indices r , and r,. It is of interest to express it in terms
of a discrete set of indices, and we can use for this
purpose the complete set of single particle states
cC/,,r,,,(r). We define then the matrix
i ) DFKSITYM A T R I C IA: ~N D NATURALORRITALS.
replaced
As is well known [S], all the physical information on
11-particle systems that involve one or at most two
body operators can be obtained with the help of one
and two particle density matrices. We shall derive
these matrices for the two electron pseudo-atom and,
in particular, discuss the correlation effects in the one
body density matrix. We shall also obtain the matrix
clements of the latter with respect to a complete set
of one particle states. The diagonalization
of the
matrix leads to the concept of natural orbital [9].
This concept gives an insight into the most effective
way 01' selecting the single particle wave functions that
appear in the approximate determinantal states for
our many body problem. Before proceeding with our As y(r,) r , ) of (2.57) is invariant under rotations,
problem we carry out the scale transformation
we see that we can write
.I
~ , l ~ / ~ n l ~ , n=
, / l nI'nlnl
r l 6 1 ~ ~1, n~2 t n I ,
on the hamiltonian (2.29) to obtain
net =
z-'
(2.59)
where
+ (z/Z) r : ] +
+ (z@)r : ] - (2 z)-' ( r l
1-1 = $[p;
+ f[p:
= $[p',l + (4 r ) ,
r i 2 ] i-
+ (4
where
cr _= :[(Z
- 2)/Z]'12 , /I
- r2)?
=p
ri2] ,
( 2 . 5 3 ~ ) in which the square bracket with index 0 indicates the
coupling of the angular momenta of the states of r l
(z;Z)"'. (2.53b) and r 2 t o zero.
This scale transformation will allow us t o discuss in a
similar fashion the expansion of the density matrix
in terms of non-interacting states (2= Z), as in
terms of the HF states (Z = Z - 1).
T h e exact two particle solution can now be written
as
Using now the relative and center of mass coordinates,
in the expression (2.57) for y(r, 1 r,), and taking
advantage of the transformation brackets introduced
in (2.21) and the value of the integrals given in (2.17),
we obtain for Y,',,,, of (2.60) the following expression
From the properties [l I] of the transformation
brackets, y!,,,, is symmetric under exchange of 112
and n , and tlii~s(as tlie pllase convention of the braI
ckets gives them in real form) 11 y,,,,,,
;I is a real symmetric matrix. The density matrix(2.59) is then broken
into real symmetric submatriccs along a diagonal
associated with the angular niomentum I. Tllcse
submatrices I! $,,,,, 1; are infinite dimensional, but
if we limit our analysis to a m a x i r n ~ ~ m
number of
quantap = 2 Ni.e. n , , 11, < h', we I~avefjnitematrices.
If we diagonalize them wc have the natural orbitals [9] as linear combinations of $,,,,,(r), cither for
non-interacting o r H F states.
Each natural orbital is associated with a definite
I
eigenvalue ol' the density submatrix I y,,,,,, / . If our
two particle state llas no corrclations, for example if
it is given by an independent particle wave function
such as H F , the eigenvalucs of the density matrix [8]
are just 1 o r 0. T o the eigenvalue I correspond then
the occupied and to 0 the empty states in the independent particle model. When there is correlation,
the eigenvalucs of the density subrnatrix differ from 1
or 0 and the extent of the difference is a good measure
of the correlation.
If we consider thc full infinite submatrix 11 i~lfr,l, 11
for definite !, its cigen\-alucs will be clearly independent
of thc basis. It' we restrict tllough ) I , , n2 < N some
bases will give a better approximation to the natural
orbital states than others. For example, if wc use tlie
non-interacting basis (i. e. z = Z ) for PLi+, calcula0
tions u p to p = 10 gi\,e the eigenvalues of 1) j)nLIIII1
that are presented in Tnble IIIa, while in 'Table lIIb
we indicate the coefficients in the expansion of the
first natural orbital i n terms of the wave functions
~ / / , , ~ ~ (In
r ) Tables 1Vu and 1Vb we carry out the
same :~nalysis starting from the H F basis (i. e.
Z = % - I). We sce that in the HF basis the eigenvalues of' li y:,,!
arc closer to 1 or 0 than is the case
I'or the non-interacting basis. Also for the first natural
orbital, thc coeficient of $,,,(r) comes closer to 1 for
the H F than for the non-interacting case. These
results again indicate that the H F basis comes closer
to the optimum single particle wave functions than
other basis.
Eigenvulzres of the densir,, rnatri,u,for P L i f us ujiritction o f t h e iliaximum number
of qlrai~tain tile IIO~I-interoctiiig
states ill tern~sof whiclz it \vas expanded
Number
of quanta
-
0
2
4
6
8
10
First
-
0.918 891
0.944 552
0.945 149
0.945 162
0.945 163
0.945 163
Second
-
0.000 299
0.000 326
0.000 328
0.000 328
0.000 328
Third
-
0.000 00
0.000 00
0.000 00
0.000 001
Fourth
Fifth
-
0.000 00
0,000 00 I
0.000 001
0.000 00
0.000 00
Coeficients of the non-interactii~ystates in the expansion of the Jirst natural orbital
of P L i f as a functiorl of'the maxintunz number of quanta in the approximation
Number
of quanta
-
0
2
4
6
8
10
Sixth
-
M. MOSHINSKY, 0.NOVARO AND A. CALLES
Eigenvalues of the density matrix for PL~' as a function of tlte maximum number
of quanta in the HF states in terilzs of which it was expanded
Number
of quanta
First
0
2
4
6
8
10
--
0.943 332
0.945 160
0.945 163
0.945 163
0.945 163
0.945 163
Second
Third
Fourth
Fifth
0.000 00
0.000 00
0.000 00
0.000 00
0.000 00
0.000 00
0.000 00
0.000 00
0.000 00
A
0.000 326
0.000 328
0.000 328
0.000 328
0.000 328
-
--
Sixth
-
0.000 00
Coeficients of the HF states in the expansion of the first natural orbital
of PLi+ as a function of the maxinzunz number of quanta in the approximation
Number
of quanta
0
2
4
6
8
10
k ) THE HYLLERAS
APPROACH.- In the original electron to the n-electron pseudo-atom with the
work of Hylleras [I51 on the helium atom, a very restriction, discussed in the introduction, that 11 < 2.
efective variational procedure was introduced in As in the two particle case, we shall limit our analysis
which the trial wave function was a product of the to the ground state of tlie pseudo-atom, and furtherground state two particle hydrogenic function of an. more consider only values of 11 in which subshells in
appropriate charge by a polynomial in the magnitude the common harmonic oscillator potential are filled. In
of the relative coordinate I r , - r2 I. It is interesting these cases the ground states will be non-degenerate
to note that in the case of the pseudo-atom this pro- with total orbital angular momentum L = 0 and
cedure is fully justified by the exact solution (2.54), spin S = 0. Besides, in the H F approximation for
these states [13, 161 we can consider that tlie single
which can be written as
particle wave functions will be those associated with
Y;(Y;, r;) = n - 3'2(4
(4 81318
spherically symmetric potentials. As discussed in
reference [16], the subsliells are filled at numbers
= ( a / ~ ) ~ ' " 1 --(a-pj
r;' +2(a-/?j2
7i4+
... }
x 7r-312(4/3)3'4 exp[-2 ~ ( r +I-:)]
:
. (2.63)
The last factor in (2.63) is just the product of two
zero quanta harmonic oscillator states of frequency 4 /I
associated with the coordinates r.,, r,, while tlie term
in the curly bracket is an infinite series in powers of
which for (a - fi) 4 1 can be approximated by a
polynomial. Thus Hylleras approach would be the
natural one to follow in the discussion of the ground
state of the two electron pseudo-atoms.
3. The n-electron pseudo-atom for the case of
closed shells. - We will now pass from the two
11 =
2 8 10 20 26 40 42 52 70 76 90 112 etc.
1 s 1 p 2 s I d 2 13 I f 3 s 2 d 1 g 3 p 2 f 1 11,
(3.1)
where underneath each value of i t we indicate the
level in the common harmonic oscillator potential that
is filled for tlie corresponding number of electrons.
In this chapter we plan to carry out the following
analysis. First in section a ) we give the exact solution
for our problem taking the result from reference [16],
but presenting it in a notation that does not use
creation operators. We then discuss in section b)
the rigorous Hartree-Fock self consistent analysis
(HFS) for the case n = 8, i. e. when the 1s - l p
shells are filled, and compare the results with those
obtained when the single particle states are of the
harmonic oscillator type. The frequency of tlie latter
will be taken from the minimization of the expectation
THE PSEUDO-ATOM : A SOLUBLE M A N Y BODY PROBLEkl
value of the hamiltonian of the pseudo-atom and we
denote the Hartree-Fock analysis using these harmonic
oscillator single particle states as the Hartree-Fock
approximation (HFA) for tliis problem. For i r = 8 the
HFS and H F A turn out to give very similar results.
In reference [16] we compared the energy and wave
functions of the exact and H F A for the 11-electron
single ionized pseudo-atoms for up t o 112 particles.
From the analysis of the 11 = 8 case we then conclude
that it is likely tliat a closely similar relation holds
between the exact and H F S energies and wave functions as the one that holds between the exact and HFA.
The comparison between the exact solution and the
approximation with antisymmetrized non-interacting
states is much poorer than between the exact and
HFA, as already seen in the two electron pseudoatom, and thus will not be discussed further here.
As in the two electron pseudo-atom, we could start
a perturbation or variation approximation using the
complete set of H F A states. We sliall see in section c)
though that tliere are some problems in the construction of this complete set for 11 > 4 tliat are related to
the antisynimetrization of the wave function. Thus tlie
analysis of chapter 2 can only be extended straiglitforwardly to the cases when 11 = 3 or 4, in which case
we d o not expect the results to be essentially different
from those of 11 = 2, and so we d o not carry out a
numerical comparison for these cases.
What is much more interesting for the 11-particle
problem is the discussion of correlation effects involving more than two particles. We sliall outline in section d) a possible procedure for studying these effects
in the case of 11 = 4 electrons with spin S = 2, tliat
correspond to a closed 1s - l p sliell. Tlie analysis
will be qualitative, as we shall leave for a future publication a quantitative discussion of correlation effects.
Finally, in section e) we discuss the form factor for
the exact ground state and compare it witli the value
one gets in the HFA. This gives us an idea of the
influence of correlation effects due to tlie interaction
on the cliarge distribution in the pseudo-atom.
C1-135
tum numbers n j l j r , i j /ij take all the possible values in
tlie filled sl~ells, for example, for 11 = 8 we have
If now we have the Iiamiltonian (1 .3) for the pseudoatom, the analysis carried in reference 1161 shows that
the exact solution for the ground state can be written as
YeX= 'I/z-,,(ri a,)[Z/(Z - I I ) ] ~ ~x'
1
xexp(-?[~"
1
-
"
( ~ - n ) " ] -11( xi = l r,)
).
As for tlie energy of tliis state, we showed in reference [16] that it lias zero quanta in the center of mass
coordinate and that the number of quanta in the
Jacobi coordinates (1.4) is the same as that in the
function !Pz-,,. The latter number we designate as
(Il(11) and as can be seen immediately from the energy
level diagram for the l~armonicoscillator it takes the
values
Thus from (1.6) the energy for these ground states
is given by
Having the exact ground state and energy, we now
pass to tlie discussion of the same in H F S starting
with 11 = 8, as the 11 = 2 case was already analyzed
in cliapter 2.
6 ) THE HARTREE-FOCI(
SELF CONSISTENT
ANALYSIS
(HFS) FOR T H E CLOSED1s - Ip SHELL.- 111tlie case of
tlie singlet ground state of the two electron pseudoatom, which corresponds to a closed 1 s shell, we
0) THE EXACT SOLUTIONFOR A CLOSED SHELL
obtained the HF integro differential equation and
PSEUDO-ATOM.
- Let us first consider an 11-electron
solved it trivially, sliowing that the K F potential is of
liamiltonian where tliere is no interaction and tlie
the harmonic oscillator type, but of frequency (Z - I)".
charge is 2,i. e. the hamiltonian has the form
The siniplificatiori in this case is due to the fact that
for the closed 1s shell tliere is no excliange term, while
the ordinary one simplifies drastically when the
interaction potential is of the harmonic oscillator
For the closed subshells indicated in (3.1) we have type. The latter property still holds for closed sliells
that the ground state is given by the determinantal beyond Is, but the excliange term no longer vanishes
and this prevents us from solving the integro differenfunction
tial equations in the simple way of section cl), chapter 2.
Yz(ri ci) = (11 !)-" det !Pnjfjmj(zx,ri) SPjn, ,
Thus for the closed 1s - I p shell and beyond, we
shall not try t o derive and solve the H F integrn
differential equation but rather follow tlie familiar
where S,,,, is the spin state uitli cri = f 4 the spin H F analysis employed in nuclear physics [13]. In this
variable, and ,uj = 1 the spin projection. Tlie quan- analysis one proposes a determinantal function of the
\I
+
1
M. MOSHINSKY, 0. NOVARO A N D A. CALLES
C4-136
type (3.3) in which tile, as Yet undetermined, single
particle state in some central potential (remember we
are dealill!? wit11 closed shells only [131), are designated by
~ , b ~ , ~ ~=
( r0,) I,
J ~2, ~..., ~ ~
(3.8)
We can develope these states in terms of harmonic
oscillator wave functions as follows :
$vitll(r)=
c:(l) $?l~!II(">
'I
(3.9)
7
where the coefficients c':(I) and the frequency Z~ are
as yet undetermined. I n the present version of the H F
analysis, the problem rests in the evaluation of Zs
and c':(l).
For the frequency 2%a simple procedure suggests
itself. Let us take the determinantal wave function (3.3) and calculate the expectation of the hamiltonian (1.3) with respect to it. Clearly this expectation
value will be a function of Z, n and 2 and if we
minimize it with respect to the latter, we get a reasonable value for the frequency 2%.This was done in
reference [16] where we showed that 2 becomes
-
=
-n
$n[R(n)
+ 3 n]-' .
.
(3.10)
Once 2 is known, we can construct a determinantal
wave function witli the single particle states (3.8, 3.9)
and take the expectation value of the liamiltonian (1.3)
with respect to it. This expectation value will give rise
to polynomials of second and fourth degree (the latter
due to the two particle interactions) in tlie coefficients c:(l). When we minimize this expression, i, e.
derive with respect to c:'(l), we get a system of algebraic
equations in the coefficients where terms in first and
third degree in tlie c':(l) appear. Solving these equations selfconsistently provides us then with the HFS
solution of our problem.
The procedure has been fully discussed in reference [13]. Here we shall apply it to the 1 s - 1 p
closed shell pseudo-atom. As in this case we have only
one state each for I = 0 and I = 1 (as in contrast
for example, to the 1 s - 1 p - 2 s shell where we
would have two I = 0 states), the index v becomes
redundant and we can suppress it c:(l). Designating
now tlie coefficients by
cn(l)
-
c:
1 = 0, 1
(3.11)
the coupled system of equations takes the form
witli
where e, and el are the eigenvalues for the energies in .Is and l p shell, while the matrix elements appearing
in the equations have the values
with
+
H(I' I" I) = [(2 1' + 1) (2 1"
I)]' x
[4n(21+ I ) ] - ~ < 1 ' 1 " 0 0 1 1 0 > ,
(3.13~)
where < ( > is a Clebsch-Gordan coefficient.
Solving the system of equation (3.12) self consistently when we go up to ten quanta, i. e. n = 0, 1, 2,
3, 4, 5, we obtain the coefficients shown in Tables 5,6
for c;f and c;. As c:, c(: are so large compared to
the others, it is quite clear that HFS solution gives
almost the same result as H F approximation (HFA)
in whicl~we take (3.3) as our determinantal wave
function with 2, given by (3.10). The analysis for
HFA was done in reference [16] and in tlie next section wc briefly summarize the results of the compari-
son between the exact and HFA for energies and wave
functions in closed shells for n up to 112.
C) COMPARISONBETWEEN THE HARTREE-FOCI\:
APPROXIMATION
(HFA) AND THE EXACTSOLUTION.
As indicated in the previous section, the HartreeFock self consistent analysis gives almost the same
result as the HFA. For the latter the wave function is
given by (3.3) with 2 determined by (3.10). In tlie
case of closed shells the energy EHFin the HFA was
discussed in reference [I61 and its explicit expression is
EllF= ( Z - lt)' [(3I(l1)
x (1
+ 3 111
+ 3 11[%(11) + + 111-'
X
( Z - it)-'
1%. (3.14)
Coeflicic~titsc z i r [lie e.vparisiori of'the 1 s HFstatc~in P F f iri terttis of'liart~iotiicoscilla~orsstates
Number
of quanta
-
0
2
4
6
8
10
Coefficients c; ill tlie expatisiori of the 1 p HF state itz PF' in tertns of hartnotiic oscillator states
offiequency Zs, asji~ticriotiqf the nzaxinium number of quanta in the approximation
Number
of quanta
-
0
2
4
6
8
10
As the exact energy and wave functions are given in
(3.7) and (3.54, we can compare them with the corresponding expressions for the HFA. This was done in
reference [16] and we just reproduce in figure 6 and 7
the difference in energies divided by E and the overlap,
1. e.
when Z - tt = 1, i. e. for single ionized pseudo-atoms
We notice from figure 6 that through the range of Z
from 3 to 113, the error in the energy ( 3 . 1 5 ~remains
)
of the order of 10 %. The error goes to zero when
i
OVERLAP
+
'
\
+
\
+ \
7 I5
F X l O 0 (RKHT SCALE)
50.
+'
/
+
+t
+
-
-
-
..
I
FIG. 7. - Overlap between the exact and H F wave functions
for single ionized closed shell pseudo-atoms of up to 112 particles.
t'
+
-
.- .. .
+
40.
+
. t
-.
..+
+
,
?I0
-
Ii
I
s
psruobr~EvEars
!1
++
9I
21. 2.7
RI' P f m o ' PSc
4i3
53
RR~PTC' P I +
~lG'P7fr.
Z -+ c ~ as
, can be seen from (3.14) and (3.7), but this
does not happen in the range of physical interest. As
for the wave functions we notice from figure 7 that the
overlap diminishes quickly as Z increases being 94
for Z = 3, but only 2 "/, when Z = 113. It is important to notice though that if we are restricted to one,
or at most two body operators, a better measure of
how good is the wave function is given by
:is
[I
FIG. 6. - Difference and deviation ratio of the H F energy
from the
value for single ionized closed sljell pseudo-atoms
of up to 1 1 2 particles.
('ffcx7
'+'F)
1 2]1/11 ,
(3.16)
which ~ ~ ~ ~ 1 1 l i Ine~ISLIrCS
;llly
the ~ v e r l i ~per
p pilrticlc.
It is clear then that the overlap (3. 1%) may be cliiitc
M . MOSHINSKY. 0. NOVARO AND A . CALLES
C4-138
small, while (3.16) remains very close to 1, if n is
large enough.
If we could construct in a systematic way all excited
states 9 5 with 2 given by (3. lo), we could start with
their help a perturbation or variation approach as was
done in sections e) and j') of chapter 2, for the two
electron pseudo-atom. Unfortunately the systematic
construction of all it-particle harmonic oscillator states
with definite spin (i. e. corresponding to definite
irreducible representations of the permutation group
of the coordinates), has only been done for 11 = 3 and 4.
Thus an analysis similar to sections e), f ) of chapter 2
can only be carried in those cases. We do no expect
though, for n = 3, 4, that the results be substantially
different from those obtained in tlie case n = 2,
and thus tlie numerical analysis will not be carried out
explicitly.
For the many electron problem there are more
interesting questions to be answered than the convergence of perturbation or variational procedures,
among which correlation effects involving more than
two particles are particularly important. We shall
address ourselves qualitatively to this question in tlie
next section.
CORRELATION
EFFECTSINVOLVING
MORETHAN
TWOPARTICLES.
THESTATEOF SPINS = 2 FOR11 = 4
PARTICLES.
- We mentioned in the previous section
that harmonic oscillator states of arbitrary spin
have been obtained in a systematic fashion for 11 = 4
particles [17]. We could then, at least in this case,
study correlations of more than two particles by
developing the exact ground state in terms of a
complete set of states of tlie HF type that belong to
the same spin value. As we like still to deal with closed
shell pseudo-atoms, to avoid tlie effect on HF solutions of possible deformed single particle potentials, it
is particularly interesting to discuss tlie state of lowest
excitation for n = 4 that has spin S = 2. This state
corresponds to a closed 1s - l p shell in which all
the spins of the particles are parallel, i. e. in configuration space the state is characterized by the irreducible representation { l 4 ) of the symmetric group
S(4).
The exact ground state for n = 4, S = 2, and the
complete set of HF states belonging to the IR { l 4 )
of S(4) have been given in reference [17]. The H F states
could be divided into sets in which only one particle,
only two, only three or all four particles are excited.
The expansion of the exact ground state in terms of HF
states belonging to the different sets, give us coefficients which provide a measure of how importai~tare
one, two, three or all four particle correlations. We
plan to discuss this problem in detail in another paper.
It is of interest also to consider for tlie n = 4,
S = 2, problem the difference of density matrices
For the HF state this difference is zero, but for the
exact wave function it gives a measure of the correlation which we also plan to discuss in the paper just
mentioned.
e) FORM
FACTORFOR THE it-ELECTRON
PROBLEM.
An important problem in atomic theory is tlie determination of the effect that the interaction between the
electrons has 011 the charge distribution in the atom,
or equivalently, on its form factor defined by
L/)
x
because u,
(3.21) by
- n - % r,',
1
We proceed to show that for the ground state (3.5),
the form factor F(q) is very easy to determine. We
notice first that the exact solution (3.5) could also be
written as
-
Ye/,, Qz -,,(rir
CT~)
- 3/4 z318
exp(-
t z'I2 rk2) ,
(3 .19)
where
The function @,-,(ri, a i ) when expressed in terms of
the Jacobi position vectors (1.4), turns out to be
independent of the center of mass coordinate r,: 1161.
Substituting (3.19) in (3.18), we can write the form
factor as
exp(in-'"
q . ri) n-)" z3I4exp(-
depends only on the relative coordinates r;,
z"'
r r ) dr,', ,
(3.21)
..., rk-1. Multiplying and dividing
and recombining terms we obtain
F(q)
=
x1
exp(iq. r l ) Y':-l,(ri,
ci)Y'z-l,(ri, G ~d)r l
... drl, x exp { -
q 2 (4 11)- [Z-
- (Z - I ~ ) - % ]1
02
=
(2i11)
xJ
exp(iq. r ) $:,,,1((~ - 11)'~ r ) $nltlt((Z -
r) d r
X
nlm
The last summation in (3.23) comes from the evaluation of the multiple integral when Y/,-,,(ri, oi) has
the determinantal form (3.3). The summation takes
place over all states in the closed shells, and (1112) is
the number of orbital states filled.
x [(z I + 1
Developing the plane wave exp(iq.r) in (3.23) into
spherical components and usingwell known results [18],
the form factor can be shown to depend on q2 only
and be given by
I:
F(~')
=
(2/11)
I)
nl
( q r ) - l (sin yr) (Z - 11)"'
R:((z
- n)'I4 r) r 2 dr] x
0
x exp { - q2(4 n)-'
[z-%- (Z - 11)-"] ) =
In (3.24) B(tr1, nl, p ) are coefficients used in the development of the radial integral in terms of simpler ones
and they have been tabulated in reference [ I I] ; the r
are gamma functions.
Had we used instead of the exact wave function (3.19) the HFA with 2 being given by (3. lo), the
form factor would have again the form (3.24) but
without the last factor and with 2 replacing Z - n.
We shall denote this approximate form factor by
FttF(q2).
For the Is - I p closed shell 11 = 8, n = 0 and
I = 0, 1, so that
I
I
IZ
In figure 8 we plot F(q2) 12, FllF(qZ) in a logarithmic scale as a function of q2 for the case of P F f
i. e. Z = 9, Z = (513). We see that the correlations
introduced by the interaction displace the curve
of F(q2) to the left of the value i t would have in
the HFA. This is to be expected as the interaction is
repulsive and so the charge density should spread
out when it is introduced. which in turn implies that
the form factor (which is the Fourier trnnsl'orm of the
charge density) should contract in, as obscrvcd. To
1
l2
FIG. 8. - From factor for P F ' , the Is - Ip closed shell
single ionized pseudo-atom. The full line gives the form factor
for the exact solution, the dotted line corresponds to the HFA,
while the dashed line applies to the state in the absence of
interaction.
corroborate this view we also give in figure 8 the form
factor for the non-interacting solution, which is
glven by (3.2%) if % is replaced by Z = 9. In this
case the lhrm I'actor extends much fr~rtheroutward
C4- 140
M. MOSHINSKY, 0 . N O V A R O AND A. CALLES
than for the exact or even the HF solution, which
means a more compact charge distribution due to the
absence of the repulsive interactions.
4. Conclusion. - Our main purpose in developing
the theory of the pseudo-atom is to provide a solvable
problem in which different approximatioil procedures
can be tested. While we did much of the testing ourselves, in connection with perturbation or variational
methods, Hartree-Fock approximations, the density
matrix approach, etc., many other procedures, in
particular those of the more modern type like correlation effects involving more than two particles, analysis of excited states, etc., remain still in the future.
I n fact, we hope that the model will prove sufficiently
interesting to entice any person introducing new
approximation techniques in atomic physics to test
them first o n the pseudo-atom.
In conclusion we would like t o quote here the well
knowns story bout the policeman o n his nightly beat,
who finds a drunkard near a lampost looking for a
quarter he had lost. H e asks him where did he lose
it and the drunkard points out somewhere in the dark.
cr Then why are ~ O L looking
I
near the lampost D, he
inquires, and the drunk replies : (( Because I have
more light here. ))
The pseudo-atom is certainly a problem in which we
have more light than in the real atom. We hope
though that Einstein was right in his phrase (< G o d
is subtle but not malicious )), and thus we could in
some way go by analytic continuation from the lamp
post to the dark spot where the real atom lies.
References
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(E. A.), Z. f: Physik 1928, 48, 469.
[2] HYLLERAS
(E. A.), Z. f: Physilc 1930, 65, 209.
[3] HARTREE
(P. A.), Proc. Cambridge Phil. Soc. 1928,
24, 89, 111, 126.
(V.), 2.f: Physilc, 1930, 61, 126.
[4] FOCK
SLATER
(J. C.), Phys. Rev., 1930, 35, 210.
[ 5 ] SINANOGLU
(O.), Proc. ROY. SOC. (London), 1961,
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[lo] MOSHINSKY
(M.), The harmonic oscillator in modern
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Gordon and Breach, N. Y.
[ I l l BRODY(T. A.) & MOSHINSKY
(M.), Tables of Transformation Brackets, 1967, Gordon and Breach,
N. Y.
[12] Reference [lo] p. 8.
[I31 Reference [lo] chapter V.
[14] TEMPLE
(G.), PYOC.Roy. Soc. (London), 1928, A 119,
176.
[IS] HYLLERAS
(E. A.), Z. f: Physik, 1929, 54, 347.
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(M.), Am. J. Pl~ys.,1970,
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(V. C.), MOSHINSKY
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[18] Reference [lo], pp. 58 and 16.